Lie Groups and Representation Theory Archives for Academic Year 2014


Organizational Meeting

When: Wed, September 4, 2013 - 2:00pm
Where: Math 1311
Speaker: () -
Abstract: Organizational meeting for Algebra and Number Theory Seminar and for Lie Group/Representation Theory Seminar.

G-valued flat deformations and local models

When: Wed, September 11, 2013 - 2:00pm
Where: Math 1311
Speaker: Brandon Levin (IAS/U. Chicago) - http://www.stanford.edu/~bwlevin/
Abstract: I will begin with a brief introduction to the deformation theory of Galois representations and its role in modularity lifting. This will motivate the study of local deformation rings and more specifically flat deformation rings. I will then discuss Kisin’s resolution of the flat deformation ring at l = p and describe conceptually the importance of local models of Shimura varieties in analyzing its geometry. Finally, I will address how to generalize these results from GL_n to a general reductive group G. If time permits, I will describe briefly the role that recent advances in p-adic Hodge theory and local models of Shimura varieties play in this situation.

Schubert varieties as variations of Hodge structure

When: Wed, September 18, 2013 - 2:00pm
Where: Math 1311
Speaker: Colleen Robles (IAS/Texas A & M) -
Abstract: Variations of Hodge structure (VHS) are constrained by a system of differential equations known as the infinitesimal period relation (IPR), or Griffiths transversality. The IPR is a distinguished homogeneous system defined on a flag variety X = G/P. I will characterize the Schubert varieties that arise as variations of Hodge structure (VHS). I will also discuss the central role that these Schubert VHS play in our study of arbitrary VHS: infinitesimally their orbits under the isotropy action `span' the space of all VHS, yielding a complete description of the infinitesimal VHS. One corollary is that they provide sharp bounds on the maximal dimension of a VHS.


Galois cohomology of real groups

When: Wed, October 9, 2013 - 2:00pm
Where: 1311.0
Speaker: Jeffrey Adams (Maryland) -
Abstract: The classification of reductive groups over a local field is stated in terms of Galois cohomology. Over R there is an alternative approach, via the Cartan involution. The equivalence of the two pictures amounts to an isomorphism between
two kinds of cohomology of the adjoint group. This generalizes to a general reductive group, and involves the notion of strong rational forms. As an application I'll compute H^1(Gal(C/R),G) for any simple simply connected group (over a p-adic field these are all trivial).

Galois cohomology of real groups II

When: Wed, October 16, 2013 - 2:00pm
Where: 1311.0
Speaker: Jeffrey Adams (Maryland) -
Abstract: See the abstract from last week's talk.

Spin conditions for local models

When: Mon, October 21, 2013 - 2:00pm
Where: Math 1311
Speaker: Brian Smithling (Johns Hopkins) - www.math.jhu.edu/~bds
Abstract: Local models are schemes which are intended to model the étale-local structure of integral models of Shimura varieties. Pappas and Zhu have recently given a general group-theoretic denition of local models with parahoric level structure, valid for any tamely ramified group, but it remains an interesting problem to characterize the local models, when possible, in terms of an explicit moduli problem. In the case of split GO(2g), Pappas and Rapoport have given a conjectural moduli description of the local model, the crucial new ingredient being what they call the _spin condition_. I will report on the proof of their conjecture in the case of a certain maximal (but not hyperspecial) parahoric level. Time permitting, I will also comment on the case of local models for ramified, quasi-split unitary groups. Here Pappas and Rapoport have also introduced a variant of the spin condition, but it turns out that this needs to be strengthened.

The Bernstein Decomposition

When: Wed, October 30, 2013 - 2:00pm
Where: Math 1311
Speaker: Jon Cohen (UMD) -
Abstract: I will explain some aspects of the Bernstein Decomposition for a reductive p-adic group. I will describe the Trace Paley-Wiener Theorem as a particular application.

Equivariant cohomology of flag manifolds

When: Wed, November 6, 2013 - 2:00pm
Where: Math 1311
Speaker: Harry Tamvakis (University of Maryland) - http://www2.math.umd.edu/~harryt/
Abstract: The equivariant cohomology ring of the flag manifold G/B for a reductive complex
Lie group G has been the subject of active research since work of Cartan and Borel
from the mid twentieth century. One has a presentation of this ring in terms of
generators and relations, coming from the characters of B, and an additive basis
of equivariant Schubert classes, coming from the Bruhat decomposition of G. A
theory of Schubert polynomials may be defined as a connection between these two
different points of view. I will discuss how this connection was recently understood
for the classical groups, and an obstruction to extending the story to the exceptional
Lie types. This talk is in part an advertisement for my graduate course, to be taught in
the spring semester.

Geometric Structure and the local Langlands conjecture

When: Wed, November 20, 2013 - 2:00pm
Where: Math 1311
Speaker: Paul Baum (Penn State) - http://www.personal.psu.edu/pxb6/
Abstract: Let G be a connected split reductive p-adic group. Examples are GL(n, F), SL(n, F), SO(n, F), Sp(2n, F), PGL(n, F) where n can be any positive integer and F can be any finite extension of the field $Q_p$ of p-adic numbers. The smooth dual of G is the set of equivalence classes of smooth irreducible representations of G. The representations are on vector spaces over the complex numbers. The smooth dual has one point for each distinct smooth irreducible representation of G. Within the smooth dual there are subsets known as the Bernstein components, and the smooth dual is the disjoint
union of the Bernstein components. This talk will explain a conjecture due to Aubert-Baum-Plymen-Solleveld (ABPS) which says that each Bernstein component is a complex affine variety. These affine varieties are explicitly identified as certain extended quotients. The infinitesimal character of Bernstein and the L-packets which appear in the local Langlands conjecture are then
described from this point of view. Granted a mild restriction on the residual characteristic of the field F over which G is defined, ABPS has been proved for any Bernstein component in the principal series of G. A corollary is that the local Langlands conjecture is valid throughout the principal series of G.
The above is joint work with Anne-Marie Aubert, Roger Plymen, and Maarten Solleveld.

The construction of the Local Langlands Correspondence of GL(2,F) using the theta correspondence

When: Mon, November 25, 2013 - 2:00pm
Where: Math 1311
Speaker: Ran Cui (UMD) -
Abstract: Fix a non-Archimedean local field F with odd characteristic, and a separable quadratic extension E. The heart of this talk is to present the construction of the Local Langlands Correspondence between G^{irr} ={equivalence classes of irreducible smooth representations of the Weil group W_F} and A^c ={equivalence classes of irreducible smooth cuspidal representations of GL(2, F)}. With the help of local class field theory, the elements in G^{irr} will be identified with elements in the set E_{reg} = {regular characters of E *}. So the question becomes: how to establish a correspondence between E_{reg} and A^c. To answer this question, Bushnell and Henniart used a modified version of the theta-correspondence and constructed the Langlands correspondence Theta <--> pi_{Theta} . Their method roughly comes down to the classical dual pair (SL(2, F);E^1), where E^1 is the set of norm 1 elements in E . However, they used the L/epsilon -factor characterization to show that pi_{Theta} is well defined. We look more closely at the dual pair construction and apply the theta-correspondence directly to (SL(2, F);E^1), and therefore obtain a correspondence theta<-->pi_{theta,psi} . By performing a series of extending and
inducing on the representations of SL(2, F), we get the corresponding representations of GL(2, F). It turns out that this approach also yields the desired Local Langlands Correspondence and moreover, a simpler proof which shows the pi_{Theta} is well defined.


seminar cancelled because of department meeting

When: Mon, December 2, 2013 - 2:00pm
Where:


Endoscopic classification of automorphic representations on classical groups: prospects and applications

When: Wed, December 4, 2013 - 4:30pm
Where: Math 3206
Speaker: Chung Pang Mok (McMaster University) - http://ms.mcmaster.ca/~cpmok/
Abstract: The recent work of Arthur, the speaker and others on the
endoscopic classification of automorphic representations of classical
groups is a landmark result in the Langlands'
program. In this talk we will outline future prospects about the
endoscopic classification and indicate some recent applications.

Local models old and new, I

When: Wed, January 29, 2014 - 2:00pm
Where: Math 1311
Speaker: Thomas Haines (UMCP) - www.math.umd.edu/~tjh
Abstract: This is an introduction to Rapoport-Zink local models for Shimura varieties with parahoric level structure. I will illustrate the general theory by concentrating on the Siegel case.



Local models old and new, II

When: Wed, February 5, 2014 - 2:00pm
Where: Math 1311
Speaker: Tom Haines (UMCP) -
Abstract: Abstract: I will explain an approach to defining local models of Shimura varieties when the level structure at $p$ is slightly deeper than Iwahori-level (the established theory, due to Rapoport-Zink, has been limited to parahoric level). I will also explain some connections to the geometric Langlands program.

The Atlas Project

When: Wed, February 12, 2014 - 2:00pm
Where: Math 1311
Speaker: Jeffrey Adams (Maryland) -
Abstract: This will be an update on the atlas project: latest developments, and a demonstration of the software.

Quantum Cohomology and Conformal Blocks divisors, I

When: Wed, February 26, 2014 - 2:00pm
Where: Math 1311
Speaker: Swarnava Mukhopadhyay (University of Maryland) -
Abstract: Conformal blocks are vector bundles on moduli space of curves with marked points that arise naturally in rational conformal field theory. Recent work on Fakhruddin has refocused our attention on conformal blocks and changed our perspective on the birational geometry of the moduli space of genus zero curves with marked points. Conformal blocks give rise to a very interesting family of numerically effective divisors and hence relate to well known conjectures on nef cones of moduli spaces of curves. I will describe joint work with Prakash Belkale and Angela Gibney where we study these divisors.
In the first talk, I plan to discuss known properties of conformal blocks divisors: in particular relation to quantum cohomology and birational geometry of the moduli space of curves with marked points. In the second talk, I will focus on our vanishing theorems, new symmetries and non-vanishing properties of these divisors (quantum cohomology of Grassmannians is one of our main tools).

Quantum Cohomology and Conformal Blocks divisors, II

When: Wed, March 5, 2014 - 2:00pm
Where: Math 1311
Speaker: Swarnava Mukhopadhyay (University of Maryland) -
Abstract: Conformal blocks are vector bundles on moduli space of curves with marked points that arise naturally in rational conformal field theory. Recent work on Fakhruddin has refocused our attention on conformal blocks and changed our perspective on the birational geometry of the moduli space of genus zero curves with marked points. Conformal blocks give rise to a very interesting family of numerically effective divisors and hence relate to well known conjectures on nef cones of moduli spaces of curves. I will describe joint work with Prakash Belkale and Angela Gibney where we study these divisors.
In the first talk, I plan to discuss known properties of conformal blocks divisors: in particular relation to quantum cohomology and birational geometry of the moduli space of curves with marked points. In the second talk, I will focus on our vanishing theorems, new symmetries and non-vanishing properties of these divisors (quantum cohomology of Grassmannians is one of our main tools).

The Arthur-Selberg trace formula and the Newton stratification of Shimura varieties

When: Mon, March 10, 2014 - 2:00pm
Where: Math 1311
Speaker: Arno Kret (IAS) -
Abstract: We show how the Arthur-Selberg trace formula can be used to study the
Newton stratification of Shimura varieties.

Cycles on modular varieties via geometric Satake

When: Wed, April 9, 2014 - 2:00pm
Where: Math 1311
Speaker: Xinwen Zhu (Northwestern) -
Abstract: I will first describe certain conjectural Tate classes in the etale cohomology of the special fibers of Shimura varieties. According to the Tate conjecture, there should exist corresponding algebraic cycles. Then I will use ideas from geometric Satake to construct these conjectural cycles. This is based on a joint work with Liang Xiao.

Unitarity of derived functor representations

When: Fri, April 25, 2014 - 2:00pm
Where: Math 1311
Speaker: Jonathan Fernandes (UMCP)

Integral models of Shimura varieties at tame primes

When: Wed, April 30, 2014 - 2:00pm
Where: Math 1311
Speaker: George Pappas (MSU) -
Abstract: Shimura varieties are important objects for arithmetic algebraic geometry
and the Langlands program. We will present some results about
integral models of some Shimura varieties at primes where the group is tamely
ramified and the level subgroup is parahoric in the sense of Bruhat-Tits.
This is joint work with M. Kisin.